The analysis of linear partial differential operators. IV: Fourier integral operators. Reprint of the 1985 original, corr. 2nd printing.

*(English)*Zbl 1178.35003
Classics in Mathematics. Berlin: Springer (ISBN 978-3-642-00117-8/pbk; 978-3-642-00136-9/ebook). vii, 352 p. (2009).

The fourth volume of the impressive monograph “The Analysis of Partial Differential Operators” by Lars Hörmander continues the detailed and unified approach of pseudo-differential and Fourier integral operators. The present book is a paperback edition of the fourth volume of this monograph. For a detailed description of the contents of the book we refer the reader to the review of the first edition [The analysis of linear partial differential operators. III: Pseudo-differential operators. Grundlehren der Mathematischen Wissenschaften, 274. Berlin etc.: Springer-Verlag (1985; Zbl 0601.35001)].

There is a longstanding connection, where symplectic geometry plays a fundamental role, between partial differential equations and geometrical and wave optics, classical and quantum mechanics. The techniques emerging from this tradition form the object of the present book: Fourier integral operators, with the study of the asymptotic properties of eigenvalues of self-adjoint elliptic operators on compact manifolds without boundary; the systematic study of the propagation of singularities, by micro-local analysis, with the Cauchy problem and pseudo-differential operators of principal type; long range scattering theory.

The book has been one of the most elegant, inspiring and influential texts in Mathematics, and contains results used in many important developments in Partial Differential Equations and Mathematical Physics. Listing these results would be a daunting task. We would only mention here that results and ideas in Chapter XXVI were used in the proof of the sufficiency of the \(\Psi\)-condition for the local solvability of principal type pseudo-differential operators [see N. Dencker, Ann. Math. (2) 163, No. 2, 405–444 (2006; Zbl 1104.35080)], the necessity of this condition having been proved by [L. Hörmander, Singularities in boundary value problems, Proc. NATO Adv. Study Inst., Maratea/Italy 1980, 69–96 (1981; Zbl 0459.35096)].

See the joint “Looking back”-review by Niels Jacob in Zbl 0712.35001.

There is a longstanding connection, where symplectic geometry plays a fundamental role, between partial differential equations and geometrical and wave optics, classical and quantum mechanics. The techniques emerging from this tradition form the object of the present book: Fourier integral operators, with the study of the asymptotic properties of eigenvalues of self-adjoint elliptic operators on compact manifolds without boundary; the systematic study of the propagation of singularities, by micro-local analysis, with the Cauchy problem and pseudo-differential operators of principal type; long range scattering theory.

The book has been one of the most elegant, inspiring and influential texts in Mathematics, and contains results used in many important developments in Partial Differential Equations and Mathematical Physics. Listing these results would be a daunting task. We would only mention here that results and ideas in Chapter XXVI were used in the proof of the sufficiency of the \(\Psi\)-condition for the local solvability of principal type pseudo-differential operators [see N. Dencker, Ann. Math. (2) 163, No. 2, 405–444 (2006; Zbl 1104.35080)], the necessity of this condition having been proved by [L. Hörmander, Singularities in boundary value problems, Proc. NATO Adv. Study Inst., Maratea/Italy 1980, 69–96 (1981; Zbl 0459.35096)].

See the joint “Looking back”-review by Niels Jacob in Zbl 0712.35001.

Reviewer: Viorel Iftimie (Bucureşti)

##### MSC:

35-02 | Research exposition (monographs, survey articles) pertaining to partial differential equations |

35Axx | General topics in partial differential equations |

35Sxx | Pseudodifferential operators and other generalizations of partial differential operators |

35P20 | Asymptotic distributions of eigenvalues in context of PDEs |

35P25 | Scattering theory for PDEs |

47G30 | Pseudodifferential operators |

47G40 | Potential operators |

58J40 | Pseudodifferential and Fourier integral operators on manifolds |

58J50 | Spectral problems; spectral geometry; scattering theory on manifolds |

53D05 | Symplectic manifolds, general |